Preparation

Total participants (NULL = did not pass captchas):

## 
##  delay normal   NULL 
##    502    502    182

Participants that passed the captchas and finished the study:

## 
##      Normal Slowed-down 
##         500         502

Descrptive Statistics

Payed bonus

Overall bonus paid out was:

## [1] 4.06

If we split bonus by whether or not participants played, we see that playing leads to a slightly higher payout:

## # A tibble: 2 x 6
##   bet_at_all mean_bonus sd_bonus se_bonus max_bonus min_bonus
## * <chr>           <dbl>    <dbl>    <dbl>     <dbl>     <dbl>
## 1 bet              4.08     3.89    0.137        72         0
## 2 no bet           4        0       0             4         4

PGSI Scores Overall

## # A tibble: 1 x 3
##    mean    sd perc_larger_7
##   <dbl> <dbl>         <dbl>
## 1  3.09  3.90         0.116
## # A tibble: 4 x 3
##   bin                       n  prob
## * <fct>                 <int> <dbl>
## 1 no risk (0)             327 0.326
## 2 low risk (1 - 2)        259 0.258
## 3 moderate risk (3 - 7)   300 0.299
## 4 problem gambler (> 7)   116 0.116

Distribution of Betting Times

We considered the time participants took for each round of roulette starting with the second round (as this was the first round to which the 1 minute slow down applied).

The first graph shows the distribution of individual betting times in seconds when restricting the shown range.

However, this plot excludes a few very long betting times, most of which occurred in the slowed down version.

## # A tibble: 19 x 3
##    ppt_id exp_cond     time
##    <chr>  <fct>       <dbl>
##  1 P0842  Slowed-down   339
##  2 P0537  Slowed-down   348
##  3 P1016  Slowed-down   306
##  4 P0537  Slowed-down  1322
##  5 P0537  Slowed-down   531
##  6 P0899  Normal        433
##  7 P0491  Slowed-down  1274
##  8 P0122  Slowed-down  3146
##  9 P0661  Slowed-down   626
## 10 P0805  Slowed-down  1075
## 11 P0070  Slowed-down   390
## 12 P0379  Slowed-down   302
## 13 P0201  Slowed-down  2561
## 14 P0787  Normal        888
## 15 P1010  Slowed-down  2322
## 16 P1177  Slowed-down   391
## 17 P1118  Slowed-down   719
## 18 P0929  Slowed-down   820
## 19 P1071  Slowed-down   352

An additional 38 betting times (of which 28 are in the slowed down version) within the range of 150 to 300 seconds are not shown here.

We can also look at some overall statistics of the betting times:

## # A tibble: 2 x 5
##   exp_cond    time_mean time_median time_sd time_IQR
## * <fct>           <dbl>       <dbl>   <dbl>    <dbl>
## 1 Normal          20.95          15   25.73       10
## 2 Slowed-down     88.63          68  144.6        15

Distribution of Proportion Bet

Our main DV clearly does not look normally distributed.

## # A tibble: 1 x 3
##   gamble_at_all gamble_everything proportion_bet_rest
##           <dbl>             <dbl>               <dbl>
## 1         0.808             0.144               0.399

Binomial confidence or credibility intervals for the probability to gamble at all:

##           method   x    n      mean     lower     upper
## 1  agresti-coull 810 1002 0.8083832 0.7828261 0.8315849
## 2     asymptotic 810 1002 0.8083832 0.7840141 0.8327523
## 3          bayes 810 1002 0.8080758 0.7835673 0.8322395
## 4        cloglog 810 1002 0.8083832 0.7826222 0.8314267
## 5          exact 810 1002 0.8083832 0.7826229 0.8323168
## 6          logit 810 1002 0.8083832 0.7828269 0.8315792
## 7         probit 810 1002 0.8083832 0.7830742 0.8317968
## 8        profile 810 1002 0.8083832 0.7832438 0.8319480
## 9            lrt 810 1002 0.8083832 0.7832510 0.8319447
## 10     prop.test 810 1002 0.8083832 0.7823333 0.8320295
## 11        wilson 810 1002 0.8083832 0.7828544 0.8315565

Numbers of Spins

The l0 variables refer to the subset of participants who made at least 1 spin (i.e., number of spins is larger than zero).

## # A tibble: 1 x 6
##    mean median    sd mean_l0 median_l10 sd_l0
##   <dbl>  <dbl> <dbl>   <dbl>      <dbl> <dbl>
## 1   4.7      2   9.2     5.8          3    10
## # A tibble: 2 x 5
##   exp_cond     mean    sd mean_l0 sd_l0
## * <fct>       <dbl> <dbl>   <dbl> <dbl>
## 1 Normal       5.96 11.7     7.39 12.6 
## 2 Slowed-down  3.41  5.66    4.20  6.02

The following shows the top 10 largest bet counts (n is the number of participants with this bet count):

## # A tibble: 69 x 3
## # Groups:   exp_cond [2]
##    exp_cond    bet_count     n
##    <fct>           <dbl> <int>
##  1 Normal            136     1
##  2 Normal            130     1
##  3 Normal             81     1
##  4 Slowed-down        62     1
##  5 Slowed-down        53     1
##  6 Normal             51     1
##  7 Normal             49     1
##  8 Normal             47     1
##  9 Normal             46     1
## 10 Normal             42     1
## # … with 59 more rows

Proportion of participants with that many spins:

## # A tibble: 2 x 6
##   exp_cond    only1 morethan1 upto2 only2 morethan2
## * <fct>       <dbl>     <dbl> <dbl> <dbl>     <dbl>
## 1 Normal      0.211     0.789 0.355 0.144     0.645
## 2 Slowed-down 0.278     0.722 0.536 0.258     0.464

Average bet size

The following table shows the average bet sizes across all bets (and not across participants):

## # A tibble: 1 x 3
##    mean    sd     n
##   <dbl> <dbl> <int>
## 1 0.727 0.628  4688
## # A tibble: 2 x 4
##   exp_cond     mean    sd     n
## * <fct>       <dbl> <dbl> <int>
## 1 Normal      0.667 0.608  2978
## 2 Slowed-down 0.832 0.648  1710

The following tables show the averages of the participant average bet sizes (i.e., averages of averages):

## # A tibble: 1 x 3
##    mean    sd     n
##   <dbl> <dbl> <int>
## 1 0.904 0.602  1002
## # A tibble: 2 x 4
##   exp_cond     mean    sd     n
## * <fct>       <dbl> <dbl> <int>
## 1 Normal      0.870 0.593   500
## 2 Slowed-down 0.937 0.610   502

Model 1: Zero-One Inflated Beta Regression on Proportion Bet

We use a custom parameterization of a zero-one-inflated beta-regression model (see also here). The likelihood of the model is given by:

\[\begin{align} f(y) &= (1 - g) & & \text{if } y = 0 \\ f(y) &= g \times e & & \text{if } y = 1 \\ f(y) &= g \times (1 - e) \times \text{Beta}(a,b) & & \text{if } y \notin \{0, 1\} \\ a &= \mu \times \phi \\ b &= (1-\mu) \times \phi \end{align}\]

Where \(1 - g\) is the zero inflation probability, zipp is \(g\) and reflects the probability to gamble, \(e\) is the conditional one-inflation probability (coi) or conditional probability to gamble everything (i.e., conditional probability to have a value of one, if one gambles), \(\mu\) is the mean of the beta distribution (Intercept), and \(\phi\) is the precision of the beta distribution (phi). As we use Stan for modelling, we need to model on the real line and need appropriate link functions. For \phi the link is log (inverse is exp()), for all other parameters it is logit (inverse is plogis()).

We fit this model and add experimental condition as a factor to the three main model parameters (i.e., only the precision parameter is fixed across conditions). The following table provides the overview of the model and all model parameters and show good convergence.

##  Family: zoib2 
##   Links: mu = logit; phi = log; zipp = logit; coi = logit 
## Formula: new_prop ~ exp_cond 
##          phi ~ 1
##          zipp ~ exp_cond
##          coi ~ exp_cond
##    Data: part2 (Number of observations: 1002) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                  -0.32      0.05    -0.42    -0.22 1.00   155141    79094
## phi_Intercept               1.28      0.05     1.18     1.37 1.00   157456    81325
## zipp_Intercept              1.42      0.11     1.21     1.65 1.00   150471    75622
## coi_Intercept              -1.61      0.13    -1.88    -1.35 1.00   158421    77535
## exp_condSloweddown         -0.17      0.07    -0.31    -0.03 1.00   158123    76631
## zipp_exp_condSloweddown     0.03      0.16    -0.28     0.35 1.00   149044    75632
## coi_exp_condSloweddown     -0.35      0.20    -0.75     0.04 1.00   153355    77331
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).

We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.

The model does not have any obvious problems, even without priors for the condition specific effects.

Posterior Predictive Checks

As expected the synthetic data generated from the model looks a lot like the actual data. This suggests that the model is adequate for the data.

Model Estimates

We first give the table showing the posterior means and 95% CIs.

## # A tibble: 6 x 8
## # Groups:   parameter [3]
##   parameter          condition   estimate .lower .upper .width .point .interval
##   <fct>              <fct>          <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Gamble at all?     Normal         0.805 0.770   0.839   0.95 mean   qi       
## 2 Gamble at all?     Slowed-down    0.810 0.775   0.843   0.95 mean   qi       
## 3 Gamble everything? Normal         0.167 0.132   0.205   0.95 mean   qi       
## 4 Gamble everything? Slowed-down    0.124 0.0935  0.157   0.95 mean   qi       
## 5 Proportion bet?    Normal         0.421 0.397   0.445   0.95 mean   qi       
## 6 Proportion bet?    Slowed-down    0.381 0.358   0.404   0.95 mean   qi

For the zero-one inflated components, we can compare the model estimates with the data. Not unsurprisingly, they match quite well.

## # A tibble: 2 x 3
##   exp_cond    gamble_at_all gamble_everything
## * <fct>               <dbl>             <dbl>
## 1 Normal              0.806             0.166
## 2 Slowed-down         0.811             0.123

The following is the main results figure on the level of condition.

Difference distribution

We can also focus and look at the difference distributions.

## # A tibble: 3 x 8
## # Groups:   parameter [3]
##   parameter          condition    estimate  .lower   .upper .width .point .interval
##   <fct>              <fct>           <dbl>   <dbl>    <dbl>  <dbl> <chr>  <chr>    
## 1 Gamble at all?     "          "  0.00472 -0.0439  0.0536    0.95 mean   qi       
## 2 Gamble everything? "          " -0.0434  -0.0921  0.00529   0.95 mean   qi       
## 3 Proportion bet?    "          " -0.0400  -0.0731 -0.00684   0.95 mean   qi

Same as a figure.

Difference Distributions with Possible Priors

The following plot shows all the difference distributions with overlayed density estimate (in black) and some possible prior distributions in colour (note again that the model did not actually include any priors). These priors are normal priors (who have a higher peak at 0 compared to Cauchy and t) with two different SDs.

Note that a prior with SD of .05 means that we expect with 95% probability that the largest effect we observe is 2.50% on the response scale.

Model 1 With PGSI

The covariate scores do not differ between conditions (evidence for the null).

## Bayes factor analysis
## --------------
## [1] Intercept only : 12.59012 ±0.02%
## 
## Against denominator:
##   pgsi ~ exp_cond 
## ---
## Bayes factor type: BFlinearModel, JZS

The following table shows mean and SDs of the covariates per group.

## # A tibble: 2 x 3
##   exp_cond    pgsi_mean pgsi_sd
## * <fct>           <dbl>   <dbl>
## 1 Normal           3.03    3.89
## 2 Slowed-down      3.15    3.91

Model with Main Effects

##  Family: zoib2 
##   Links: mu = logit; phi = log; zipp = logit; coi = logit 
## Formula: new_prop ~ exp_cond + pgsi_c 
##          phi ~ 1
##          zipp ~ exp_cond + pgsi_c
##          coi ~ exp_cond + pgsi_c
##    Data: part2 (Number of observations: 1002) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                  -0.32      0.05    -0.42    -0.22 1.00   202255    76647
## phi_Intercept               1.28      0.05     1.19     1.38 1.00   199280    79155
## zipp_Intercept              1.45      0.11     1.23     1.68 1.00   197844    77180
## coi_Intercept              -1.65      0.14    -1.92    -1.38 1.00   195737    78392
## exp_condSloweddown         -0.17      0.07    -0.31    -0.03 1.00   203440    75284
## pgsi_c                      0.02      0.01     0.01     0.04 1.00   200229    77371
## zipp_exp_condSloweddown     0.02      0.16    -0.29     0.34 1.00   194430    75775
## zipp_pgsi_c                 0.06      0.02     0.02     0.11 1.00   189734    78791
## coi_exp_condSloweddown     -0.38      0.20    -0.78     0.02 1.00   196127    76236
## coi_pgsi_c                  0.08      0.02     0.03     0.12 1.00   183586    78854
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).

We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.

## # A tibble: 3 x 8
## # Groups:   parameter [3]
##   parameter          condition    estimate  .lower   .upper .width .point .interval
##   <fct>              <fct>           <dbl>   <dbl>    <dbl>  <dbl> <chr>  <chr>    
## 1 Gamble at all?     "          "  0.00368 -0.0449  0.0520    0.95 mean   qi       
## 2 Gamble everything? "          " -0.0448  -0.0923  0.00229   0.95 mean   qi       
## 3 Proportion bet?    "          " -0.0412  -0.0739 -0.00793   0.95 mean   qi

Plot of Relationships and BANOVA

The figure below provides an alternative visualisation of the relationships between covariates and betting behaviour. In particular, participants were categorized into one of three experimental betting behavior groups: participants who did not bet at all (“none”, 14% of participants), participants who bet some of their money (68% of participants), and participants who bet “all” of their money (18%). For both gambling scales we see a positive relationship between the betting behavior group and the gambling score. Participants who bet more have on average higher scores on the two gambling scales.

## 
##      none      some       all 
## 0.1916168 0.6916168 0.1167665

This is supported by Bayesian ANOVAs with Bayes factors of over 100 for the effect of betting behavior group on the PGSI scores. However, this effect was not moderated by gambling speed condition. In particular, there was evidence for the absence of both a main effect of gambling message condition and an interaction of gambling message condition with betting behaviour group for both gambling scale scores (Bayes factors for the null > 25).

## Bayes factor analysis
## --------------
## [1] exp_cond                                    : 0.07942733 ±0.02%
## [2] gamble_cat                                  : 146.7407   ±0.03%
## [3] exp_cond + gamble_cat                       : 12.85686   ±2.8%
## [4] exp_cond + gamble_cat + exp_cond:gamble_cat : 0.6547955  ±2.34%
## 
## Against denominator:
##   Intercept only 
## ---
## Bayes factor type: BFlinearModel, JZS
##             denominator
## numerator    exp_cond gamble_cat exp_cond + gamble_cat exp_cond + gamble_cat + exp_cond:gamble_cat
##   gamble_cat 1847.483          1              11.41341                                    224.1015

Model With Interactions

##  Family: zoib2 
##   Links: mu = logit; phi = log; zipp = logit; coi = logit 
## Formula: new_prop ~ exp_cond * pgsi_c 
##          phi ~ 1
##          zipp ~ exp_cond * pgsi_c
##          coi ~ exp_cond * pgsi_c
##    Data: part2 (Number of observations: 1002) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                         -0.32      0.05    -0.42    -0.22 1.00   143528    78188
## phi_Intercept                      1.28      0.05     1.19     1.38 1.00   139972    78368
## zipp_Intercept                     1.44      0.12     1.22     1.67 1.00   134911    76222
## coi_Intercept                     -1.64      0.14    -1.92    -1.38 1.00   154460    76195
## exp_condSloweddown                -0.17      0.07    -0.31    -0.03 1.00   152425    76075
## pgsi_c                             0.03      0.01    -0.00     0.05 1.00    93780    78211
## exp_condSloweddown:pgsi_c         -0.01      0.02    -0.04     0.03 1.00    92641    79992
## zipp_exp_condSloweddown            0.05      0.16    -0.28     0.37 1.00   124973    78255
## zipp_pgsi_c                        0.05      0.03    -0.01     0.12 1.00    95088    75259
## zipp_exp_condSloweddown:pgsi_c     0.04      0.05    -0.06     0.13 1.00    93369    73923
## coi_exp_condSloweddown            -0.40      0.21    -0.81     0.01 1.00   129111    79579
## coi_pgsi_c                         0.07      0.03     0.01     0.13 1.00    95090    74977
## coi_exp_condSloweddown:pgsi_c      0.02      0.04    -0.07     0.11 1.00    93523    77862
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).

We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.

## # A tibble: 3 x 8
## # Groups:   parameter [3]
##   parameter          condition    estimate  .lower   .upper .width .point .interval
##   <fct>              <fct>           <dbl>   <dbl>    <dbl>  <dbl> <chr>  <chr>    
## 1 Gamble at all?     "          "  0.00687 -0.0425  0.0563    0.95 mean   qi       
## 2 Gamble everything? "          " -0.0470  -0.0956  0.00154   0.95 mean   qi       
## 3 Proportion bet?    "          " -0.0412  -0.0742 -0.00818   0.95 mean   qi

Model 2: Total Number of Spins

##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: bet_count | trunc(lb = 1) ~ exp_cond 
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              1.03      0.34     0.32     1.44 1.00    23993    16478
## exp_condSloweddown    -0.80      0.13    -1.05    -0.56 1.00    43513    42460
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     0.18      0.06     0.07     0.29 1.00    23346    16223
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

The data seems to be well described by the model.

When we zoom in (i.e., ignore data points above 50 for the plot), we can see the that the real and synthetic data match quite well.

## # A tibble: 2 x 7
##   exp_cond     mean .lower .upper .width .point .interval
##   <fct>       <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Normal       2.94  1.37    4.22   0.95 median qi       
## 2 Slowed down  1.32  0.602   1.94   0.95 median qi

## # A tibble: 1 x 7
##   exp_cond              mean .lower .upper .width .point .interval
##   <chr>                <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Slowed down - Normal -1.59  -2.51 -0.713   0.95 median qi
## # A tibble: 1 x 2
##   exp_cond             prop_larger_0
## * <chr>                        <dbl>
## 1 Slowed down - Normal             0

Model 2 With PGSI

With Main Effects

##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: bet_count | trunc(lb = 1) ~ exp_cond + pgsi_c 
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              1.01      0.39     0.25     1.44 1.00    30638    18810
## exp_condSloweddown    -0.80      0.13    -1.05    -0.56 1.00    62474    56482
## pgsi_c                 0.00      0.02    -0.03     0.03 1.00    66789    55930
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     0.17      0.06     0.06     0.29 1.00    29076    18493
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

## # A tibble: 2 x 7
##   exp_cond     mean .lower .upper .width .point .interval
##   <fct>       <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Normal       2.91  1.29    4.21   0.95 median qi       
## 2 Slowed down  1.31  0.557   1.93   0.95 median qi

With Interactions

##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: bet_count | trunc(lb = 1) ~ exp_cond * pgsi_c 
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                     1.00      0.37     0.23     1.43 1.00    30654    18625
## exp_condSloweddown           -0.80      0.13    -1.06    -0.56 1.00    63806    60174
## pgsi_c                        0.00      0.02    -0.04     0.05 1.00    53370    54458
## exp_condSloweddown:pgsi_c    -0.01      0.03    -0.07     0.06 1.00    53112    55471
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     0.17      0.06     0.06     0.28 1.00    29265    18160
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

## NOTE: Results may be misleading due to involvement in interactions

## NOTE: Results may be misleading due to involvement in interactions
## # A tibble: 2 x 7
##   exp_cond     mean .lower .upper .width .point .interval
##   <fct>       <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Normal       2.88  1.26    4.19   0.95 median qi       
## 2 Slowed down  1.29  0.548   1.92   0.95 median qi
## NOTE: Results may be misleading due to involvement in interactions

Model 3: Average Bet Size

The following shows the distribution of average bet sizes for those that bet at least once. The left plot shows the actual averages and the right plot the average scaled to the 0 to 1 range (i.e., after dividing by 2 as pre-registered). We can see that the distribution of actual averages is multi-modal with peaks at several prominent numbers: the first peak at the minimum possible bet size 0.1, the second peak at 0.5, the largest peak at the center at 1 and finally a large peak at the maximum of 2. As pre-registered we will analyse the data with a one-inflated beta-regression model.

##  Family: oib 
##   Links: mu = logit; phi = log; oi = logit 
## Formula: average_scaled ~ exp_cond 
##          phi ~ 1
##          oi ~ exp_cond
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                -0.52      0.05    -0.62    -0.42 1.00   125539    81492
## phi_Intercept             1.23      0.05     1.14     1.33 1.00   113749    79510
## oi_Intercept             -2.05      0.16    -2.36    -1.75 1.00   107921    74429
## exp_condSloweddown        0.06      0.07    -0.08     0.20 1.00   129652    80618
## oi_exp_condSloweddown     0.21      0.21    -0.20     0.63 1.00   113596    77606
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

When comparing the actual data with synthetic data from the model, we can see an obvious problem due to the use of prominent numbers. Whereas the peak at 1 is nicely recovered by the model, the other peaks are not. However, the overall shape of the synthetic data is at least similar to the shape of the observed data.

## Joining, by = c("exp_cond", ".chain", ".iteration", ".draw")

## # A tibble: 4 x 8
## # Groups:   parameter [2]
##   parameter     exp_cond    value .lower .upper .width .point .interval
##   <fct>         <fct>       <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Mean bet size Normal      0.373 0.350   0.396   0.95 median qi       
## 2 Mean bet size Slowed-down 0.387 0.363   0.411   0.95 median qi       
## 3 Bet maximum   Normal      0.114 0.0860  0.148   0.95 median qi       
## 4 Bet maximum   Slowed-down 0.138 0.107   0.174   0.95 median qi

Difference distribution:

## # A tibble: 2 x 8
## # Groups:   parameter [2]
##   parameter     exp_cond              value  .lower .upper .width .point .interval
##   <fct>         <chr>                 <dbl>   <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Mean bet size Slowed-down - Normal 0.0140 -0.0188 0.0467   0.95 median qi       
## 2 Bet maximum   Slowed-down - Normal 0.0236 -0.0223 0.0693   0.95 median qi

Model 3 with PGSI

With Main Effect

##  Family: oib 
##   Links: mu = logit; phi = log; oi = logit 
## Formula: average_scaled ~ exp_cond + pgsi_c 
##          phi ~ 1
##          oi ~ exp_cond
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                -0.52      0.05    -0.62    -0.43 1.00   131605    79448
## phi_Intercept             1.25      0.05     1.16     1.34 1.00   129109    79536
## oi_Intercept             -2.05      0.16    -2.37    -1.75 1.00   130545    74257
## exp_condSloweddown        0.06      0.07    -0.08     0.20 1.00   130068    75254
## pgsi_c                    0.03      0.01     0.01     0.05 1.00   138523    79400
## oi_exp_condSloweddown     0.21      0.21    -0.20     0.64 1.00   134356    79885
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

## Joining, by = c("exp_cond", ".chain", ".iteration", ".draw")

Difference distribution:

With Interaction

##  Family: oib 
##   Links: mu = logit; phi = log; oi = logit 
## Formula: average_scaled ~ exp_cond * pgsi_c 
##          phi ~ 1
##          oi ~ exp_cond
##    Data: part_nozero (Number of observations: 810) 
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
##          total post-warmup samples = 1e+05
## 
## Population-Level Effects: 
##                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept                    -0.52      0.05    -0.62    -0.43 1.00   152532    74735
## phi_Intercept                 1.25      0.05     1.16     1.35 1.00   145226    75484
## oi_Intercept                 -2.05      0.16    -2.36    -1.75 1.00   149134    73545
## exp_condSloweddown            0.06      0.07    -0.08     0.20 1.00   159353    75933
## pgsi_c                        0.02      0.01    -0.01     0.04 1.00    98907    81465
## exp_condSloweddown:pgsi_c     0.03      0.02    -0.01     0.06 1.00    98626    81467
## oi_exp_condSloweddown         0.21      0.21    -0.20     0.63 1.00   148880    77368
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Difference distribution:

Riskiness of Bets

Do either of the spin speeds affect the riskiness of bets chosen in the roulette? Different bets come with different potential payoffs in roulette. If £1 is bet, then this can provide a total payoff of between £2 (e.g., bets on red or black) and £36 (bets on a single number). These numbers, 2 and 36, are the decimal odds for these two bets, representing the total potential payoff Additionally, gamblers can place multiple bets per spin, (e.g., betting £0.50 on red and £0.50 on 7). Since the number 7 is a red colour, the bet on 7 can only win if the first bet on red also wins. Multiple bets per spin can either be placed in a way that accentuates risk, as in this example, or in a way that hedges risk (for example, a bet on black added to the bet on 7, or betting on reds and blacks together). The purpose of this exploratory analysis is to see if either warning label affects risk taking.

Following our pre-registration we have measures the amount of risk taken by looking at the variance of the decimal odds for each number in the roulette table. It will measure the concentration of the bet, with more risk represented by more concentrated bets (e.g., betting on a single number), and lower risk represented by more spread bets (e.g., betting on reds, betting on evens). For example, if a participant places a bet on number 7, the decimal odds for that number is 36 (36 times the amount bet if the roulette stops on the number 7). If a participant places a bet on red, then every red number on the table gets a decimal odd of 2 (the participant will win twice the amount bet if the roulette stops on any red number). In order to calculate the proposed risk variable, we will also assign decimal odds of zero for the numbers in which participants did not bet (since they win zero times the amount bet if the roulette stops on those numbers). In the two examples above, every number except 7 will be assigned zero, and every non-red number will be assigned zero, respectively. The risk variable will be the variance of the array of decimal odds for every number on the roulette table, taking into account the bets the participant has placed, and including zeroes for non-winning numbers. The value of this risk variable can range between 0.03 and 35.03. The value cannot be zero in our task because participants are not able to bet the same amount across all numbers including the zero due to the limitations in the values of the available tokens and total bet amounts. Higher numbers indicate more risk taking (with the highest, 35.03, associated with concentrating the bet on a single number). Lower numbers indicate lower risk taking (with the lowest, 0.03, associated with spreading the bet across every red and black number, excluding zero).

The following plot shows the distribution of this variable after aggregating within participants. The left plot shows the original variable on the scale from 0.03 to 35.03. The middle plot shows the variable after dividing by 36 so that the variable ranges from just above 0 to just below 1. The right plot shows the variable after subtracting 0.03 and dividing by 35 so that the smallest value is mapped to 0 and the largest value is mapped to 1.

Beta Regression

We first analyse the data with a beta-regression model. For this, the data needs to be between 0 and 1, but exclude exactly 0 and 1. Consequently, we use the transformation shown above in the middle panel. As before, we fit the data and allow for an effect of gambling message.

The model does not show any obvious problems. In addition, we can see that the 95%-CIs for the slowed down effect includes 0.

##  Family: beta 
##   Links: mu = logit; phi = log 
## Formula: average_odds2 ~ exp_cond 
##          phi ~ 1
##    Data: bets_av (Number of observations: 810) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept             -2.06      0.06    -2.17    -1.95 1.00     2631     2563
## phi_Intercept          1.59      0.06     1.48     1.69 1.00     2497     2575
## exp_condSloweddown    -0.04      0.07    -0.18     0.10 1.00     2914     2692
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

The posterior predictive distribution shows some problems, but at least the shape of the synthetic data is similar to the distribution of the actual data (posterior predictive distributions of a model assuming a Gaussian response distribution was considerably worse and is therefore not included here).

The following tables show the average riskiness per condition and differences from the no warning message group on the fitted scale, i.e., the (0, 1) scale used for the beta regression.

## # A tibble: 2 x 7
##   condition  estimate .lower .upper .width .point .interval
##   <chr>         <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 Normal        0.113 0.102   0.125   0.95 mean   qi       
## 2 SlowedDown    0.110 0.0992  0.121   0.95 mean   qi
## # A tibble: 1 x 7
##   condition           estimate  .lower .upper .width .point .interval
##   <fct>                  <dbl>   <dbl>  <dbl>  <dbl> <chr>  <chr>    
## 1 SlowedDown - Normal -0.00388 -0.0180 0.0102   0.95 mean   qi

Session Info

## R version 4.0.3 (2020-10-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.5 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C               LC_TIME=en_GB.UTF-8       
##  [4] LC_COLLATE=en_GB.UTF-8     LC_MONETARY=en_GB.UTF-8    LC_MESSAGES=en_GB.UTF-8   
##  [7] LC_PAPER=en_GB.UTF-8       LC_NAME=C                  LC_ADDRESS=C              
## [10] LC_TELEPHONE=C             LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] emmeans_1.5.4          binom_1.1-1            BayesFactor_0.9.12-4.2 Matrix_1.3-2          
##  [5] coda_0.19-4            tidybayes_2.3.1        brms_2.14.4            Rcpp_1.0.6            
##  [9] forcats_0.5.1          stringr_1.4.0          dplyr_1.0.4            purrr_0.3.4           
## [13] readr_1.4.0            tidyr_1.1.2            tibble_3.1.0           ggplot2_3.3.3         
## [17] tidyverse_1.3.0       
## 
## loaded via a namespace (and not attached):
##   [1] readxl_1.3.1         backports_1.2.1      plyr_1.8.6           igraph_1.2.6        
##   [5] splines_4.0.3        svUnit_1.0.3         crosstalk_1.1.1      rstantools_2.1.1    
##   [9] inline_0.3.17        digest_0.6.27        htmltools_0.5.1.1    rsconnect_0.8.16    
##  [13] fansi_0.4.2          magrittr_2.0.1       modelr_0.1.8         RcppParallel_5.0.3  
##  [17] matrixStats_0.58.0   xts_0.12.1           prettyunits_1.1.1    colorspace_2.0-0    
##  [21] rvest_0.3.6          ggdist_2.4.0         haven_2.3.1          xfun_0.21           
##  [25] callr_3.5.1          crayon_1.4.1         jsonlite_1.7.2       lme4_1.1-26         
##  [29] zoo_1.8-8            glue_1.4.2           gtable_0.3.0         MatrixModels_0.4-1  
##  [33] V8_3.4.0             distributional_0.2.2 pkgbuild_1.2.0       rstan_2.21.2        
##  [37] abind_1.4-5          scales_1.1.1         mvtnorm_1.1-1        DBI_1.1.1           
##  [41] miniUI_0.1.1.1       xtable_1.8-4         stats4_4.0.3         StanHeaders_2.21.0-7
##  [45] DT_0.17              htmlwidgets_1.5.3    httr_1.4.2           threejs_0.3.3       
##  [49] arrayhelpers_1.1-0   ellipsis_0.3.1       pkgconfig_2.0.3      loo_2.4.1           
##  [53] farver_2.0.3         sass_0.3.1           dbplyr_2.1.0         utf8_1.1.4          
##  [57] labeling_0.4.2       tidyselect_1.1.0     rlang_0.4.10         reshape2_1.4.4      
##  [61] later_1.1.0.1        munsell_0.5.0        cellranger_1.1.0     tools_4.0.3         
##  [65] cli_2.3.1            generics_0.1.0       broom_0.7.5          ggridges_0.5.3      
##  [69] evaluate_0.14        fastmap_1.1.0        yaml_2.2.1           processx_3.4.5      
##  [73] knitr_1.31           fs_1.5.0             pbapply_1.4-3        nlme_3.1-152        
##  [77] mime_0.10            projpred_2.0.2       xml2_1.3.2           compiler_4.0.3      
##  [81] bayesplot_1.8.0      shinythemes_1.2.0    rstudioapi_0.13      gamm4_0.2-6         
##  [85] curl_4.3             reprex_1.0.0         statmod_1.4.35       bslib_0.2.4         
##  [89] stringi_1.5.3        highr_0.8            ps_1.5.0             Brobdingnag_1.2-6   
##  [93] lattice_0.20-41      nloptr_1.2.2.2       markdown_1.1         shinyjs_2.0.0       
##  [97] vctrs_0.3.6          pillar_1.5.0         lifecycle_1.0.0      jquerylib_0.1.3     
## [101] bridgesampling_1.0-0 estimability_1.3     cowplot_1.1.1        httpuv_1.5.5        
## [105] R6_2.5.0             promises_1.2.0.1     gridExtra_2.3        codetools_0.2-18    
## [109] boot_1.3-27          colourpicker_1.1.0   MASS_7.3-53.1        gtools_3.8.2        
## [113] assertthat_0.2.1     withr_2.4.1          shinystan_2.5.0      mgcv_1.8-34         
## [117] parallel_4.0.3       hms_1.0.0            grid_4.0.3           minqa_1.2.4         
## [121] rmarkdown_2.7        shiny_1.6.0          lubridate_1.7.9.2    base64enc_0.1-3     
## [125] dygraphs_1.1.1.6